Q. 105.0( 1 Vote )

# Mark the tick against the correct answer in the following:Let S be the set of all real numbers and let R be a relation on S, defined by a R b ⇔ |a – b| ≤ 1. Then, R isA. reflexive and symmetric but not transitiveB. reflexive and transitive but not symmetricC. symmetric and transitive but not reflexiveD. an equivalence relation

According to the question ,

Given set S = {…….,-2,-1,0,1,2 …..}

And R = {(a, b) : a,b S and |a – b| ≤ 1 }

Formula

For a relation R in set A

Reflexive

The relation is reflexive if (a , a) R for every a A

Symmetric

The relation is Symmetric if (a , b) R , then (b , a) R

Transitive

Relation is Transitive if (a , b) R & (b , c) R , then (a , c) R

Equivalence

If the relation is reflexive , symmetric and transitive , it is an equivalence relation.

Check for reflexive

Consider , (a,a)

|a – a| ≤ 1 and which is always true.

Ex_if a=2

|2-2| ≤ 1 0 ≤ 1 which is true.

Therefore , R is reflexive ……. (1)

Check for symmetric

a R b |a – b| ≤ 1

b R a |b – a| ≤ 1

Both can be true.

Ex _ If a=2 and b=1

|2 – 1| ≤ 1 is true and |1–2| ≤ 1 which is also true.

Therefore , R is symmetric ……. (2)

Check for transitive

a R b |a – b| ≤ 1

b R c |b – c| ≤ 1

|a – c| ≤ 1 will not always be true

Ex _a=-5 , b= -6 and c= -7

|6-5| ≤ 1 , |7 – 6| ≤ 1 are true But |7 – 5| ≤ 1 is false.

Therefore , R is not transitive ……. (3)

Now , according to the equations (1) , (2) , (3)

Correct option will be (A)

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